The aim of the present paper is to study some properties of $\alpha$-paracompact sets of a topological space which is not Hausdorff or regular. It will be shown that some properties of $\alpha$-paracompact sets of a Hausdorff or regular space are valid although the topological space is not Hausdorff or regular. Notation is standard except that $\alpha(A)$ will be used to denote the interior of the closure of $A$.